If , are cardinals, define to be the cardinal number of the set of

If α, β are cardinals, define α^β to be the cardinal number of the set of all functions B → A, where A,B are sets such that IAI = α, IBI = β

(a) α^β is independent of the choice of A,B.

(b) α^β+ϒ = (α^β}(α^ϒ); (αβ)^ϒ = (α^ϒ)(β^ϒ); α^βϒ = (α^β)^ϒ

(c) If α ≤ β, then α^ϒ ≤β^ϒ.

(d) If α,β are finite with α > 1, β > l and ϒ is infinite, then α^ϒ = β^ϒ.

(e) For every finite cardinal n, αⁿ = α.α··· α (n factors). Hence αⁿ = α if α. is infinite.

(f) If P(A) is the power set of a set A, then IP(A)I = 2^IAI.